Question

$ABCD$ is a quadrilateral in which $P,Q,R$ and $S$ are the mid-points of the sides $AB,BC,CD$ and $DA$. $AC$ is a diagonal. Show that:
i) $SR\parallel AC$ and $SR=\frac{1}{2}AC$
ii) $PQ=SR$
iii) $PQRS$ is a parallelogram.

Answer 1

From the given figure, let us consider $\Delta ADC$ of which, $S$ and $R$ are the mid-points of the sides $AD$ and $CD$

According to the mid point theorem we get,

$SR\parallel AC$ and $SR=\frac{1}{2}AC$ ...... $\left(1\right)$

Similar things happen for $\Delta ABC$, from which we get

$PQ\parallel AC$ and $PQ=\frac{1}{2}AC$ ...... $\left(2\right)$

From $\left(1\right)$ and $\left(2\right)$ we get,

$PQ=SR$ and $PQ\parallel SR$

$\Rightarrow PQRS$ is a parallelogram.